Critical Damping

Ask

For the control system in the given figure, the value of K for critical damping is

(a) 1

(b) 5.125

(c) 6.831

(d) 10



Answer is ( b )

Concept

We should first consider simplifying the block - diagram in the question as:



Combining the two cascaded blocks , we get : $G(s) = (K)(\frac{2}{s^2+7s+2})$

A closed loop control system having a form as below:



Thus , the above diagram simplifies to $\frac{C(s)}{R(s)} = \frac{(\frac{2K}{s^2+7s+2})}{1+(\frac{2K}{s^2+7s+2})}$

$\implies \frac{C(s)}{R(s)}=  \frac{2K}{s^2+7s+2+2K}$

Now to Solve the Above Equation we need to understand the characteristic equation of a second - order system.

In general a second - order control system equation is given by :
$$\frac{C(s)}{R(s)}= \frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2}$$
where,
$\zeta$ = damping ratio ; $\omega_n$ = natural frequency
The characteristic Equation of the system is given by :$s^2+2\zeta\omega_n s+\omega_n^2 = 0$
Comparing this equation to an algebraic equation of the form $ax^2+bx+c=0$
The roots of the above equation are $ x= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
For a critical damping case $ b^2 - 4ac= 0 , \implies x= \frac{-b}{2a}$

The characteristic equation for the given problem : $s^2+7s+2+2K = 0 $
As for critical damping case $ b^2 - 4ac= 0 $
$\implies  7^2 - 4(1)(2+2K)= 0 $
$\implies  K=41/8 = 5.125 $